Question: The sum of two angles is $82^\circ$. Angle 2 is $153^\circ$ smaller than $4$ times angle 1. What are the measures of the two angles in degrees?
Solution: Let $x$ equal the measure of angle 1 and $y$ equal the measure of angle 2. The system of equations is then: ${x+y = 82}$ ${y = 4x-153}$ Since we already have solved for $y$ in terms of $x$ , we can use substitution to solve for $x$ and $y$ Substitute ${4x-153}$ for $y$ in the first equation. ${x + }{(4x-153)}{= 82}$ Simplify and solve for $x$ $ x+4x - 153 = 82 $ $ 5x-153 = 82 $ $ 5x = 235 $ $ x = \dfrac{235}{5} $ ${x = 47}$ Now that you know ${x = 47}$ , plug it back into $ {y = 4x-153}$ to find $y$ ${y = 4}{(47)}{ - 153}$ $y = 188 - 153$ ${y = 35}$ You can also plug ${x = 47}$ into $ {x+y = 82}$ and get the same answer for $y$ ${(47)}{ + y = 82}$ ${y = 35}$ The measure of angle 1 is $47^\circ$ and the measure of angle 2 is $35^\circ$.